Testing Hypotheses
In Exercises 13–24, assume that a sim...

Question

Testing Hypotheses

In Exercises 13–24, assume that a simple random sample has been selected and test the given claim. Unless specified by your instructor, use either the P-value method or the critical value method for testing hypotheses. Identify the null and alternative hypotheses, test statistic, P-value (or range of P-values), or critical value(s), and state the final conclusion that addresses the original claim.

Lead in Medicine Listed below are the lead concentrations (in ) measured in different Ayurveda medicines. Ayurveda is a traditional medical system commonly used in India. The lead concentrations listed here are from medicines manufactured in the United States (based on data from “Lead, Mercury, and Arsenic in US and Indian Manufactured Ayurvedic Medicines Sold via the Internet,” by Saper et al., Journal of the American Medical Association, Vol. 300, No. 8). Use a 0.05 significance level to test the claim that the mean lead concentration for all such medicines is less than 14 μg/g.

3.0 6.5 6.0 5.5 20.5 7.5 12 20.5 11.5 17.5

Accepted Answer

Welcome back, everyone. In this problem, a researcher is investigating the caffeine content per serving in milligrams of a certain brand of energy drinks. The following caffeine concentrations were measured in a simple random sample 82, 95, 88, 76, 110, 98, 85, 105, 90, and 100 mg. Test the claim at the 0.05 significance level that the mean caffeine content for all such energy drinks is less than 100 mg, assuming that the population is normally distributed. Now notice here in this problem we are testing a claim about a population being with the standard deviation for the population sigma not known. Now the sample is a simple random sample, and even though our sample size is less than 30, we are told to assume that the population is normally distributed. So since sigma is unknown. OK, the population standard deviation. And our sample size is less than 30, we can use the student's T test to test our claim. Let's start by stating our hypothesis. Now we know on our in our problem we are testing the claim that the caffeine content is less than 100 mg. So another hypothesis is that it's not true. In other words, that the mean is greater than or equal to 100 mg, which means that our alternative hypothesis is that the mean is less than 100 mg, OK? So if we were to visualize what we're really looking for here, we're assuming. Given this normal distribution, OK. That the mean value, OK, let's say that here this is a mean value of 100, and we're testing that this mean value of 100 is less than 100. In other words, if we get our T value based on our significance level and the degrees of freedom, OK. Then we are trying to figure out if the test statistic will be in the shaded red region. If it is, in other words, if it's less than our critical value here. Then that means we would reject the null hypothesis, and if it is greater than or equal to the test to the critical value, then we would have to accept the null hypothesis, OK? So let's find the, the critical value, the critical T value for this significance level and the degrees of freedom, and then let's compare it to the test statistic to see where the test statistic falls. And of course, because we're testing that it's less than 100 mg, then we're going to use a left tailed test. For a left tilt test. OK. With the degrees of freedom equal to the sample size minus 1, so that's 10 minus 1, which is 9, OK, and a significance level alpha of 0.05. Using the table of critical values, the T value, OK, is going to be approximately equal to 1.833, and that's the value that we found in our table. And now remember, this is a left tail test so we can negate that value. So our critical value is -1.833. Now, we can compute a test statistic to see if it is less than this value or greater than or equal to it. Now, how do we compute the test, the test statistic? Well, recall, OK, let me put that in black here, recall. That the test statistic T will be equal to. The sample mean minus the mean from our hypotheses divided by the population sample, the standard deviation sample or the sample standard deviation divided by the square root of the sample size N. OK. Now, in this case, we need to figure out our sample mean and standard deviation. OK. So now, we know that our sample mean will be equal to the sum of all the values. Divided by the number of values. So in this case, we know that our sum of values is 929. I'll leave you to add them, but that's what you should find. And the number of values is 10. So that means our sample mean is 92.9, OK? Next, we also know that the sample standard deviation will be equal to the square root of the sum of the squared mean differences, OK, divided by the sample size minus 1. So in that case here, It's going to be equal to the square root of 1/9 of those squared sample squared differences, OK? And I'll leave you to solve again this on your own in the interest of time, but you should find that you get the sample standard deviation to be 10.64007101 or approximately 10.640, OK? So now in this case. Since we have the sample mean, let me highlight that in red. Since we have the sample mean, the sample standard deviation, and we know that there are 10 values, so N equals 10, then substituting that into the test statistic formula, that means it will be equal to 92.9 minus 100. Divided by 10.640 divided by the square root of 10. And if we go ahead and solve, we should get it to be approximately equal to -2.110. This is our computed test statistic. Let's go back to our diagram. What does this mean? Well, since we know the critical value was 1.1.833, and we found that the test statistic falls at -2.110, that means. That it is within the rejection region. Therefore, we can say that at the 0.05 significance level, OK? It's important to make note of what we find. Then there is sufficient evidence. We rejected the no hypothesis, so there is sufficient evidence. To support the claim. So we have enough evidence to support the claim that the mean caffeine content for all such energy drinks, energy drinks is less than 100 mg. Thanks a lot for watching everyone. I hope this video helped.

Key Concepts

Hypothesis Testing

P-value

Significance Level

Watch next